Problem 12. Two squares are 20 square meters and 125 square meters. Do their sides have a common measure?

20 : 125 = 4 : 25, upon dividing each term by 5.The squares are therefore in the same ratio as square numbers. Therefore their sides have a common measure.

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Now, what will be the case if two squares are not in the same ratio as two square numbers? What if one square is twice the size of another?

2 and 1 are not both square numbers. What must we say about the ratio of the sides? The answer is the subject of the next Lesson.

The theorem of this Lesson is credited to Theaetatus ("The-AY-ti-tus") of Athens, who was born around 417 B.C. There is a story recorded by Plato in his dialogue Theaetatus, that when Theaetatus was a boy, his teacher Theodorus brought him to meet Socrates. It was then that Socrates suggested to the youngster that it is not necessary to be intelligent, only to want to know.